EE 369 POWER SYSTEM ANALYSIS Lecture 7 Transmission Line Models Tom Overbye and Ross Baldick

Announcements For lecture 7 to 10 read Chapters 5 and 3. HW 6 is problems 5.2, 5.4, 5.7, 5.9, 5.14, 5.16, 5.19, 5.26, 5.31, 5.32, 5.33, 5.36; case study questions chapter 5 a, b, c, d, is due Thursday, 10/15. Homework 7 is 5.8, 5.15, 5.17, 5.24, 5.27, 5.28, 5.29, 5.34, 5.37, 5.38, 5.43, 5.45; due 10/22.

Transmission Line Models Previous lectures have covered how to calculate the distributed series inductance, shunt capacitance, and series resistance of transmission lines: That is, we have calculated the inductance L, capacitance C, and resistance r per unit length, We can also think of the shunt conductance g per unit length, Each infinitesimal length dx of transmission line consists of a series impedance rdx + jωLdx and a shunt admittance gdx + jωCdx, In this section we will use these distributed parameters to develop the transmission line models used in power system analysis.

Transmission Line Equivalent Circuit Our model of an infinitesimal length of transmission line is shown below: dx L Units on z and y are per unit length!

Derivation of V, I Relationships dx L

Setting up a Second Order Equation

V, I Relationships, cont’d

Equation for Voltage

Real Hyperbolic Functions

Complex Hyperbolic Functions

Determining Line Voltage

Determining Line Voltage, cont’d

Determining Line Current

Transmission Line Example

Transmission Line Example, cont’d

Transmission Line Example, cont’d Squares and crosses show real and reactive power flow, where a positive value of flow means flow to the left. Receiving end Sending end

Lossless Transmission Lines

Lossless Transmission Lines

Lossless Transmission Lines If load power P > SIL then line consumes VArs; otherwise, the line generates VArs.

Transmission Matrix Model Often we are only interested in the terminal characteristics of the transmission line. Therefore we can model it as a “black box:” VS VR + - IS IR Transmission Line

Transmission Matrix Model, cont’d

Equivalent Circuit Model To do this, we’ll use the T matrix values to derive the parameters Z' and Y' that match the behavior of the equivalent circuit to that of the T matrix. We do this by first finding the relationship between sending and receiving end for the equivalent circuit.

Equivalent Circuit Parameters

Equivalent circuit parameters

Simplified Parameters

Simplified Parameters

Three Line Models The long line model is always correct. The other models are usually good approximations for the conditions described.

Power Transfer in Short Lines Often we'd like to know the maximum power that could be transferred through a short transmission line V1 V2 + - I1 I2 Transmission Line with Impedance Z S12 S21

Power Transfer in Lossless Lines

Limits Affecting Max. Power Transfer Thermal limits limit is due to heating of conductor and hence depends heavily on ambient conditions. For many lines, sagging is the limiting constraint. Newer conductors/materials limit can limit sag. Trees grow, and will eventually hit lines if they are planted under the line, Note that thermal limit is different to the steady-state stability limit that we just calculated: Thermal limits due to losses, Steady-state stability limit applies even for lossless line!

Tree Trimming: Before

Tree Trimming: After

Other Limits Affecting Power Transfer Angle limits while the maximum power transfer (steady-state stability limit) occurs when the line angle difference is 90 degrees, actual limit is substantially less due to interaction of multiple lines in the system Voltage stability limits as power transfers increases, reactive losses increase as I2X. As reactive power increases the voltage falls, resulting in a potentially cascading voltage collapse.